Feynman Path Integral Formulation

3.2 Perturbatively Non-renormalizable Theories: The Sigma Model 75
unambiguous), then the calculable constant relating m to g in Eq. (3.29) uniquely
determines the coefficient a 0 in Eq. (3.22). For example, in the large N limit the
value for a 0 will be given later in Eq. (3.60).
In general one can write down the complete renormalization group equations
for the cutoff-dependent n-point functions Γ (n) (p i ,g,h,Λ) (Brezin and Zinn-Justin,
1976; Zinn-Justin, 2002). For this purpose one needs to define the renormalized
truncated n-point function Γ (n)
r ,
Γ (n)
r (p i ,g r ,h r , μ)=Z n/2 (Λ/μ,g)Γ (n) (p i ,g,h,Λ) , (3.30)
where μ is a renormalization scale, and the constants g r , h r and Z are defined by
g =(Λ/μ) d−2 Z g g r π(x)=Z 1/2 π r (x)
h = Z h h r Z h = Z g / √ Z . (3.31)
The requirement that the renormalized n-point function Γ r
(n) be independent of the
cutoff Λ then implies
[
Λ ∂
∂Λ + β(g) ∂
∂g − n 2 ζ (g)+ρ(g)h ∂ ]
Γ (n) (p i ,g,h,Λ)=0 , (3.32)
∂h
with the renormalization group functions β(g), ζ (g) and ρ(g) defined as
Λ ∂
∂Λ | ren.fixed g = β(g)
Λ ∂
∂Λ | ren.fixed (−lnZ) =ζ (g)
2 − d + 1 2 ζ (g)+β(g) g
= ρ(g) . (3.33)
Here the derivatives of the bare coupling g,oftheπ-field wave function renormalization
constant Z and of the external field h with respect to the cutoff Λ are evaluated
at fixed renormalized (or effective) coupling, at the renormalization scale μ.
To determine the renormalization group functions β(g), ζ (g) and ρ(g) one can
in fact follow a related but equivalent procedure, in which, instead of requiring the
to be independent of the cutoff Λ at fixed renormalization
scale μ as in Eq. (3.33), one imposes that the bare n-point functions
Γ (n) be independent of the renormalization scale μ at fixed cutoff Λ. One can show
(Brezin, Le Guillou and Zinn-Justin, 1976) that the resulting renormalization group
functions are identical to the previous ones, and that one can obtain the scale dependence
of the couplings [i.e. β(g)] either way. Physically the latter way of thinking
is perhaps more suited to a situation where one is dealing with a finite cutoff theory,
where the ultraviolet cutoff Λ is fixed and one wants to investigate the scale
(momentum) dependence of the couplings, for example g(k 2 ).
renormalized n-point functions Γ (n)
r